© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a357 K11a359

Knotscape

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DrawMorseLink

PD Presentation:

X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21

Gauss Code:

{1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -6, 11, -5}

DT (Dowker-Thistlethwaite) Code:

8 14 16 18 22 20 2 4 6 12 10

Alexander Polynomial:

3t-3 - 5t-2 + 5t-1 - 5 + 5t - 5t2 + 3t3

Conway Polynomial:

1 + 12z2 + 13z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{31, 6}

Jones Polynomial:

q3 - q4 + 2q5 - 3q6 + 4q7 - 4q8 + 5q9 - 4q10 + 3q11 - 2q12 + q13 - q14

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q10 + q14 + 2q24 + q26 + q28 + q30 - q36 - q38 - q40 - q42

HOMFLY-PT Polynomial:

- 3a-12 - 4a-12z2 - a-12z4 + 3a-10 + 7a-10z2 + 5a-10z4 + a-10z6 + 3a-8z2 + 4a-8z4 + a-8z6 + a-6 + 6a-6z2 + 5a-6z4 + a-6z6

Kauffman Polynomial:

3a-17z - 4a-17z3 + a-17z5 + a-16z2 - 3a-16z4 + a-16z6 - a-15z + 2a-15z3 - 3a-15z5 + a-15z7 - 3a-14z2 + 6a-14z4 - 4a-14z6 + a-14z8 + a-13z - 4a-13z3 + 9a-13z5 - 5a-13z7 + a-13z9 - 3a-12 + 18a-12z2 - 26a-12z4 + 20a-12z6 - 7a-12z8 + a-12z10 + 5a-11z - 20a-11z3 + 26a-11z5 - 12a-11z7 + 2a-11z9 - 3a-10 + 14a-10z2 - 26a-10z4 + 20a-10z6 - 7a-10z8 + a-10z10 - 7a-9z3 + 9a-9z5 - 5a-9z7 + a-9z9 - 2a-8z2 + 4a-8z4 - 4a-8z6 + a-8z8 + 3a-7z3 - 4a-7z5 + a-7z7 - a-6 + 6a-6z2 - 5a-6z4 + a-6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{12, 41}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=6 is the signature of 11358. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                         j = 25                   2 1   j = 23                 1       j = 21               3 2       j = 19             2 1         j = 17           2 3           j = 15         2 2             j = 13       1 2               j = 11     1 2                 j = 9     1                   j = 7 1 1                     j = 5 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 358]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 358]]
Out[3]=	PD[X[8, 2, 9, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[22, 10, 1, 9], X[20, 12, 21, 11], X[2, 14, 3, 13], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 358]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 358]]
Out[5]=	DTCode[8, 14, 16, 18, 22, 20, 2, 4, 6, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 358]][t]
Out[6]=	3 5 5 2 3 -5 + -- - -- + - + 5 t - 5 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 358]][z]
Out[7]=	2 4 6 1 + 12 z + 13 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 358]}
In[9]:=	{KnotDet[Knot[11, Alternating, 358]], KnotSignature[Knot[11, Alternating, 358]]}
Out[9]=	{31, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 358]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 13 14 q - q + 2 q - 3 q + 4 q - 4 q + 5 q - 4 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 358]}
In[12]:=	A2Invariant[Knot[11, Alternating, 358]][q]
Out[12]=	10 14 24 26 28 30 36 38 40 42 q + q + 2 q + q + q + q - q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 358]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -3 3 -6 4 z 7 z 3 z 6 z z 5 z 4 z 5 z z --- + --- + a - ---- + ---- + ---- + ---- - --- + ---- + ---- + ---- + --- + 12 10 12 10 8 6 12 10 8 6 10 a a a a a a a a a a a 6 6 z z > -- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 358]][a, z]
Out[14]=	2 2 2 2 2 -3 3 -6 3 z z z 5 z z 3 z 18 z 14 z 2 z --- - --- - a + --- - --- + --- + --- + --- - ---- + ----- + ----- - ---- + 12 10 17 15 13 11 16 14 12 10 8 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 6 z 4 z 2 z 4 z 20 z 7 z 3 z 3 z 6 z 26 z > ---- - ---- + ---- - ---- - ----- - ---- + ---- - ---- + ---- - ----- - 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 26 z 4 z 5 z z 3 z 9 z 26 z 9 z 4 z z > ----- + ---- - ---- + --- - ---- + ---- + ----- + ---- - ---- + --- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 8 4 z 20 z 20 z 4 z z z 5 z 12 z 5 z z z > ---- + ----- + ----- - ---- + -- + --- - ---- - ----- - ---- + -- + --- - 14 12 10 8 6 15 13 11 9 7 14 a a a a a a a a a a a 8 8 8 9 9 9 10 10 7 z 7 z z z 2 z z z z > ---- - ---- + -- + --- + ---- + -- + --- + --- 12 10 8 13 11 9 12 10 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 358]], Vassiliev[3][Knot[11, Alternating, 358]]}
Out[15]=	{12, 41}
In[16]:=	Kh[Knot[11, Alternating, 358]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 15 4 q + q + q t + q t + q t + 2 q t + q t + 2 q t + 2 q t + 15 5 17 5 17 6 19 6 19 7 21 7 21 8 > 2 q t + 2 q t + 3 q t + 2 q t + q t + 3 q t + 2 q t + 23 8 25 9 25 10 29 11 > q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a358

K11a357 K11a359